Obstructive sleep-disordered breathing (SDB) affects at least 2-3% of children and is a frequent reason for adenotonsillectomy, the second most common surgical procedure performed in childhood. Among consequences of SDB, neurobehavioral problems such as inattention, hyperactivity, and daytime sleepiness are some of the most important and have been linked to levels of SDB that would otherwise be considered mild. Although adenotonsillectomy was once thought highly effective, recent studies in the midst of epidemic obesity show that more than half of operated patients still have residual SDB, though usually at mild levels, after surgery. Many sleep specialists now advocate for routine evaluation, by polysomnography or otherwise, after SDB is treated by adenotonsillectomy. However, this is far from usual practice, in part because evaluation results that would merit further treatment remain poorly defined, and the potential clinical benefit of continuous positive airway pressure (CPAP), the mainstay of treatment for adult SDB, has not been well documented in children who have undergone adenotonsillectomy. The main goals of the research now proposed, therefore, are to examine neurobehavioral benefits of CPAP after adenotonsillectomy for SDB, and to determine what role a well-validated symptom-based questionnaire and polysomnography can play in post- operative determination of which patients may benefit from CPAP. Four months after adenotonsillectomy, 120 children will be assessed with sleep laboratory-based polysomnography, sleep questionnaires, Multiple Sleep Latency Tests of daytime sleepiness, standard parental behavioral ratings, and neuropsychological testing. Results will show that after adenotonsillectomy, an SDB questionnaire can help identify children who should have polysomnography to assess for residual SDB, and that residual SDB is associated with persistent neurobehavioral morbidity. The investigators will then randomize 80 subjects to receive CPAP and 40 to receive no CPAP, treat the first group and monitor adherence, and after 6 months re-assess all 120 participants with parental sleep questionnaires, polysomnography, Multiple Sleep Latency Tests, parent behavioral ratings, and neuropsychological testing. Results will document that polysomnography identifies children who benefit from further treatment for SDB; implicate residual SDB after adenotonsillectomy as a cause of persistent neurobehavioral problems; and suggest that successful treatment of residual SDB can provide tangible clinical benefits. The investigators have a long and successful track record of clinical research on childhood SDB; novel approaches to its subjective and objective assessment; and neurobehavioral benefits of its treatment. Results of the critical, timely, and innovative protocol now proposed are likely to have substantial impact: they will help to inform common and consequential clinical decisions, guide allocation of costly resources, justify subsequent large multicenter randomized clinical trials, and improve the health, cognition, and behavior of many children.